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Math Help - Bergman Space Question

  1. #1
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    Bergman Space Question

    Let G=\{z \in \field{C} :0<|z|<1\} show that every f \in L_a^2(G) has a removable singularity at z=0.

    I'm quite certain this is an easy question, but I'm lacking the background for the appropriate proof. Any help?
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  2. #2
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    Since no one apparently knows this, let me go ahead and type this:

    z=0 is a removable singularity of f \in L_a^2(G) if there exists a holomorphic function g definied on G \cup \{0\} such that f(z)=g(z), \forall z \in G . Hence let  g=f, \forall f \in G. f is holomorphic on G \cup \{0\} , since f is holomorphic on G.

    Am I missing something? It seems that by definition of being holomorphic on G, you're holomorphic at {0}, no?
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  3. #3
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    Quote Originally Posted by MattMan View Post
    Let G=\{z \in \field{C} :0<|z|<1\} show that every f \in L_a^2(G) has a removable singularity at z=0.

    I'm quite certain this is an easy question, but I'm lacking the background for the appropriate proof. Any help?
    This is a nice problem, unfortunately I don't have a full solution yet.

    It's easy to see that f can't have a pole (because then f(z)=\frac{g(z)}{z^m} with g(0)\neq 0 and \frac{1}{z^m} \in L^2(G) iff m\leq 0), but I haven't been able to deduce the case where f has an essential singularity (It's informally clear since f takes on (almost) the whole plane an infinite number of times in any neighbourhood of 0, but the proof eludes me)
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